multiuser_waterfilling

8/11/2019 multiuser_waterfilling

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Multiuser Water-lling in the Presence of Crosstalk Wei Yu

Department of Electrical and Computer EngineeringUniversity of Toronto

Toronto, Ontario M5S 3G4, CanadaEmail: [email protected]

Abstract Spectrum optimization is an important part of thedesign of interference-limited multiuser communication systems.While traditional water-lling provides a closed-form solutionto the transmit optimization problem in a single-user system,the multiuser version of the problem often leads to nonconvexproblem formulations which are difcult to solve. Motivated byhigh-speed parallel-link and digital subscriber line applications,this paper investigates two practical multiuser settings in whichglobal or local optimal solutions to the multiuser spectrumoptimization problems can be found efciently. The rst part of this paper considers a high-speed transmission system in which

practical (but suboptimal) minimum-mean-squared-error linearequalizers (MMSE-LEs) are used at the receiver. The optimalsingle-user transmit spectrum in this case involves a modiedwater-lling solution. Surprisingly, such a modied water-llingspectrum can be shown to be near-optimal in a multiusersetting as well, if the direct-link and the crosstalk characteristicsare symmetric and if crosstalk is reasonably small. Thus, forpractical parallel-link systems using MMSE linear equalizers, theoptimal single-user and multiuser spectra are nearly identical.The second part of this paper considers numerical techniquesfor solving a nonconvex multiuser rate maximization problemfor digital subscriber line applications. A new ingredient in theproposed approach is a taxation scheme that takes into accountthe effect of interference between neighboring lines. This leadsto a modied iterative water-lling algorithm which is capable

of nding local optimum solutions to the multiuser spectrumoptimization problem efciently.

I. INTRODUCTION

Spectrum optimization is an important aspect of multiusercommunication system design. While for traditional single-user systems, spectrum optimization amounts to an optimalchoice of bandwidth, power and modulation format for themaximization of single-user capacity, the design of multiusersystems also has to account for the effect of interference.Mathematically, while single-user spectrum optimization prob-lem often leads to convex problem formulations which areanalytically tractable, its multiuser counterpart often leads tononconvex problems which are numerically difcult to solve.This paper makes progress on multiuser spectrum optimizationproblems by focusing on two specic scenarios in whichefcient numerical algorithms exist. In the rst part of thepaper, we show that the formulation of transmit spectrumoptimization problem depends on receiver design. While theuse of an optimal receiver leads to a well-known water-llingsolution, the use of a less-complex minimum-mean-squared-error (MMSE) linear receiver leads to a different water-llingspectrum. Surprisingly, this spectrum is near-optimal in some

multiuser setting as well. In the second half of this paper, wefocus on efcient numerical solutions to a nonconvex mul-tiuser rate maximization problem in the presence of crosstalk interference. We show that a modied iterative water-llingalgorithm can be used to arrive at local optimal solutions of the problem efciently.

We are motivated to study the multiuser transmit spectrumoptimization problem by high-speed parallel transmission lineand digital subscriber line applications. (See Fig. 1.) In high-

speed backplane communication systems, data transmissionsoccur on parallel links. The crosstalk between the links is amajor transmission impairment. In these high-speed applica-tions, the information-theoretical optimal multi-carrier modu-lation or decision-feedback equalization may be too complexto implement; suboptimal linear equalizers are often used. Therst part of this paper tackles the optimal transmit powerallocation problem in the presence of crosstalk when MMSElinear equalizers are used at the receiver.

The second part of the paper is motivated by applicationsin which information-theoretic achievable rate is the correctmetric to optimize. This is the case for digital subscriber linesystems where optimal receivers are used. The multiuser ratemaximization problem for an interference channel is a well-known nonconvex optimization problem. This paper providesnumerical insights to the problem and proposes an efcientalgorithm that aims to reach local optimal solutions of theproblem.

II . VARIATIONS OF WATER -FILLING

Traditional water-lling refers to the shaping of the transmitspectrum for capacity maximization in a single-transmittersingle-receiver Gaussian channel with additive colored noise.The water-lling solution is derived from a maximum mutual

information argument; it is widely used in frequency-selectiveor intersymbol interference (ISI) channels, as well as in fadingchannels with adaptive modulation.

A key assumption in the derivation of water-lling spectrumis the use of an optimal receiver. However, when a suboptimalreceiver such as an MMSE linear equalizer (MMSE-LE) isused, the water-lling spectrum needs to be suitably modied.In the following, we begin by reviewing the structures of optimal water-lling spectra for both optimal and suboptimalreceivers.


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Tx

Tx

Tx

Rx

Rx

Rx

Fig. 1. A multiuser system in the presence of crosstalk. The optimal transmitspectrum depends on the receiver structure.

A. Water-lling for Capacity Maximization

The traditional justication for water-lling is based onmutual information maximization. In an ISI channel with achannel transfer function H (f ), the maximum achievable datarate subject to an input power constraint P is the solution tothe following optimization problem:

maxmize

F

0log 1 +

|W (f )|2 |H (f )|2

2df

subject to F

0|W (f )|2 P

|W (f )|2 0, (1)

where the optimizing variable is input power spectral density,denoted as |W (f )|2 , 2 is the additive white Gaussian noisepower level, and F is the maximum frequency range used. Thisoptimization problem is convex; it has the following simplesolution:

|W (f )|2 = K 2

|H (f )|2

+

, (2)

where K is a constant chosen to satisfy the total power

constraint. Equation (2) is referred to as the water-lling powerallocation because 2

| H ( f ) | 2 can be thoughts of as the bottomof a bowl, |W (f )|2 as the water lling the bowl, and K asthe water level.

In formulating the optimization problem (1) via mutualinformation, it is implicitly assumed that an optimal equalizeris used at the receiver 1. This is true, for example, in anorthogonal frequency-division multiplex (OFDM) implemen-tation where H (f ) is simply the channel gain and the optimalW (f ) is simply the power spectrum across the frequencytones. This can also true for single-carrier systems. As single-carrier systems with decision-feedback equalizers can also becapacity achieving [1], the solution W (f ) obtained from (1)is the optimal transmit spectrum in this case as well. However,when the optimal transmit spectrum has a multi-band structure,multiple decision-feedback equalizers may have to be used.

B. Water-lling with MMSE Linear Equalizer

Clearly, water-lling is no longer the optimal transmitspectrum when suboptimal receivers are used. This is the case,

1In addition, (1) also assumes that use of capacity-achieving codes. How-ever, the use of practical codes can be easily taken into account by includingan SNR gap term in (1).

010 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 0 0 0 0

111111111111111111111111111111111111111111111111111111111111

0 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 00 0 0 0 0 0 0

1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

K

K|H (f )|

f

f

|W (f )| 2

|W (f )| 2

2

|H (f )| 2

2

|H (f )| 2

Fig. 2. Capacity-maximizing water-lling vs. MMSE-LE water-lling.

for example, when a linear equalizer is used. Consider theuse of an MMSE-LE. Fixing the transmit spectrum W (f ),an MMSE-LE minimizes the total mean-squared error at thereceiver. An interesting question is then: What is the optimaltransmitter for such a suboptimal receiver? This problem hasbeen formulated and solved in [2], where an analytic solutionis shown to exist. We derive the solution below for the sakeof completeness.

It is convenient to begin the derivation in matrix form.Consider an ISI channel with a Toeplitz matrix channel H :

y = H x + z , (3)

where S zz = 2I . The transmit spectrum can be representedby a Toeplitz matrix W in

x = W u . (4)

Without loss of generality, we may set S uu = I . The inputpower constraint then becomes tr( W W T ) P . The MMSElinear equalizer L has the following form:

L = S uy S 1yy= W T H T (HW W T H T + 2 I ) 1 (5)

The minimum mean-squared error in this case is:

MMSE = I W T H T (HW W T H T + 2 I ) 1HW = ( I + W T H T 2 HW ) 1 (6)


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We now formulate the MMSE minimization problem subjectto a power constraint. This is most easily done in the frequencydomain. As both H and W are assumed to be Toeplitz, we nowuse their frequency-domain representations H (f ) and W (f ),respectively. In this case, the design problem becomes:

minimize

F

0

11 + 2 |W (f )|2 |H (f )|2

df

subject to F

0|W (f )|2 P

|W (f )|2 0, (7)

Interestingly, this problem also has an analytical solution.Following a Lagrangian approach, it is not difcult to showthat the optimal transmit spectrum is [2]:

|W (f )|2 = K|H (f )|

2

|H (f )|2

+

(8)

Comparing this solution with the capacity-maximizing water-lling solution (2), it is interesting to note that the two are very

similar, except that the water-lling level in (8) is no longer aconstant. Instead, the MMSE-LE water-lling level is scaledby the inverse of the square root of the effective channel gain.Fig. 2 illustrates the difference between capacity-maximizingand MMSE-LE water-lling.

The capacity-maximizing water-lling solution always putsmore power in high-SNR frequency bands in such a way that|W (f )|2 +

2

| H ( f ) | 2 is a constant. The MMSE-LE water-llingsolution is less aggressive in spectrum shaping. The MMSE-LE water-lling process effectively depresses the water-llinglevel in the high-SNR frequency bands, and raises the water-lling level in the low-SNR frequency bands. The MMSE-LEwater-lling process is essentially a combination of channel

inversion and water-lling .

III . M ULTIUSER WATER -FILLING WITH MMSE-LE

The main objective of this paper is to derive the optimalmultiuser water-lling solution in the presence of crosstalk. Asmentioned earlier, in many high-speed multiline transmissionsystems, crosstalk interference between the lines is a dominantsource of channel noise. Thus, the optimal transmit spectrumhas to consider not only the maximization of each lines ownrate (or the minimization of its own MSE), but also the effectof crosstalk interference each line emits to its neighboringlines. In this section, we consider the multiuser water-llingproblem assuming MMSE-LE receivers.

In general, multiuser spectrum optimization often leads tononconvex problem formulations, which are difcult to solve.In this section, we focus on a special, yet very practical,subclass of such problems for which an analytical solutionis possible. This specic class includes parallel transmissionline systems in which the direct channel transfer functions areidentical for all lines and each line receives a same amount of crosstalk. This is the case, for example, for parallel backplanetransmission line systems, where a set of equal-length linesare in close proximity to each other. The crosstalk channels

are symmetric, and each line (except the rst and the last linein the backplane) experiences the same amount of crosstalk.This is the scenario depicted in Fig. 1.

The optimal multiuser spectrum optimization problem inthese scenarios simplies signicantly because of the sym-metry. It is reasonable to assume that every transmitter andreceiver must operate in the same way. In particular, thetransmit power spectral density for each line must be identical.In addition, linear equalizers are assumed to be used at thereceiver. We now ask: What is the optimal transmit spectrumwith an MMSE-LE receiver?

By symmetric, the channel models for all users are identicaland it simplies to the following:

y = H x + Gv + z , (9)

where v is the crosstalk signal and G is the Toeplitz crosstalk channel. By assumption, v has the same power spectrum asx . As before, H is the Toeplitz matrix representing the directchannel. Let the transmit lter be:

x = W u (10)

The total noise power spectrum then becomes GW W T GT +2I . Following the same derivation as in the single-user case,the minimum mean-squared error with an MMSE-LE is:

MMSE = ( I + W T H T (GWW T GT + 2I ) 1HW ) 1 . (11)

Again, the minimization of the MMSE subject to powerconstraint is most easily done in the frequency domain. As H ,W and G are all Toeplitz, the design problem then becomes:

minimize F

0

11 + |W ( f ) |

2 | H ( f ) | 2

| G ( f ) | 2 | W ( f ) | 2 + 2df

subject to F

0 |W (f )|2

P |W (f )|2 0, (12)

where H (f ), W (f ) and G(f ) are the respective frequency-domain representations. The main result of this section is thatthe above problem has almost the same solution as in thesingle-user case.

Theorem 1: The optimal transmit spectrum that minimizesthe mean-squared error in a symmetric crosstalk environmentwith MMSE linear equalizers at the receiver is

|W (f )|2 = |H (f )|2

|H (f )|2 + |G(f )|2 K|H (f )|

2

|H (f )|2

+

(13)where H (f ) and G(f ) represent the direct and the totalcrosstalk channel gains, 2 is the additive background noiselevel, and K is a constant chosen to satisfy the total powerconstraint.

Proof: The proof involves a manipulation of the Karush-Kuhn-Tucker condition for the optimization problem (12).Form the Lagrangian of (12) and take its derivation, we arriveat (14) at the top of the next page. After cancelation of terms,we get the optimality condition (13).


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|G(f )|2(( |G(f )|2 + |H (f )|2)|W (f )|2 + 2) (|G(f )|2 + |H (f )|2)( |G(f )|2 |W (f )|2 + 2 )(( |G(f )|+ |H (f )|2)|W (f )|2 + 2)2

= K (14)

The most remarkable aspect of this result is that the optimaltransmit spectrum in a multiuser system with MMSE-LE isalmost exactly the same as that of a single-user system withMMSE-LE, except by a scaling factor. The scaling factor is

| H ( f ) | 2

| H ( f ) | 2 + | G ( f ) | 2 . In many practical systems of interests, thecrosstalk channel gains are always much smaller than thedirect channel gains. In these cases, the scaling factor is almost1 and the optimal multiuser spectrum and the optimal single-user spectrum are almost identical.

This is a surprising result, as the scaling factor is inde-pendent of the noise power level. One would expect that insituations where the crosstalk channel is much weaker thanthe direct channel, but where the background noise is evensmaller, multiuser spectrum optimization would have produceda different transmit spectrum as compared to the single-user

case. Yet, the two turn out to be near identical, even thoughthe justications for the single-user and the multiuser casesare very different.

IV. M ULTIUSER CAPACITY M AXIMIZATION

The second part of this paper deals with a rate maximizationproblem for multiuser systems in the presence of crosstalk.Here, we assume that optimal receivers are used. In this case,the rate maximization problem becomes the maximization of mutual information in an interference environment.

Although the capacity region of the interference channel isstill an open problem, in practical systems where the crosstalk channels are typically weaker than the direct channel, treating

interference as noise is often the most practical communicationstrategy. This is the case for digital subscriber line (DSL)applications, which is the motivating example in this part of the paper.

When the crosstalk interference is regarded as noise, a K -user spectrum optimization problem for rate maximizationbecomes:

maxK

k =1

kN

n =1log 1 +

P k (n)|H kk (n)|2

2 + j = k P j (n)|H jk (n)|2

s.t .N

n =1P k (n) P k k = 1 , , K

P k (n) 0 k = 1 , , K (15)Here, the objective function is a weighted rate maximizationproblem with weighting factor k . We discretize the entirefrequency range into N tones, corresponding to that of apractical OFDM modulation system, where H jk (n) denotesthe channel transfer function from user j to user k in tone n ,and P k (n) denotes the power spectrum for user k in tone n.The total power constraint for user k is denoted as P k . In aDSL application, the crosstalk channels cannot be assumed tobe symmetric (i.e. H ij (n) = H ji (n) in general.)

The multiuser rate maximization problem (15) has beenconsidered extensively in the literature. An early idea, callediterative water-lling [3], calls for the use a greedy algorithmto arrive at an approximate solution to this problem. In theiterative water-lling algorithm, each user iteratively performsa single-user water-lling, while regarding interference asnoise. As an update of each users power spectrum changes theinterference pattern for all other users, the water-lling processneeds to be performed iteratively until the power spectra of allusers converge. While efcient and very easy to implement,the iterative water-lling algorithm does not attempt to solvethe weighted rate maximization problem optimally. Each usermaximizes its own rate only, with no regard to other usersrates.

Fundamentally, the weighted rate maximization problem

is a difcult problem to solve numerically, because of thenonconvex nature of the optimization objective. The bestnumerical algorithm to date is known as the optimal spectrumbalancing (OSB) algorithm [4], which relies on a dualizationof the power constraint using a Lagrangian multiplier and anexhaustive search over each users power spectrum for eachxed value of Lagrangian multipliers. The dual optimizationapproach works despite nonconvexity, because the spectrumoptimization problem is separable in the frequency domainand it satises an unusual property that its duality gap is zeroin the limit of large N [5]. Nevertheless, because the optimalspectrum balancing algorithm involves an exhaustive searchover the user power, the complexity of OSB is exponential

in K , which makes it impractical for systems with a largenumber of users.The OSB algorithm depends crucially on the separability of

the rate expression over frequency. Separability in this contextmeans that frequency tones are independent of each other, andthe power of each user in tone n affects the aggregate rateonly in tone n. While separability holds for (15), the rateexpression of (15) is only an idealization. In practical OFDMsystems where the received OFDM symbol from the directchannel and that from the crosstalk channel do not perfectlyalign with each other, intercarrier interference (ICI) results [6].In this case, the total crosstalk interference for user k in tonen has the following expression [6]:

Crosstalk =j = k

N 2 1

m = N 2

(m)P j (n m)|H jk (n m)|2

(16)where (0) = 1 and (m) = 2 / (N 2 sin2(m/N )) , for m =1, is the tone-to-tone coupling coefcient 2 [6]. In this case,

2 Note that due to the circulant structure of OFDM modulation, the toneindices in H ij (n ) and P j (n ) in (16) should be regarded as circulant, e.g.P j ( 1) = P j (N 1) .


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01

0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 k k + t k(n )

n

|P k(n )| 2

I k(n ) + 2

|H kk (n )| 2

Fig. 3. Water-lling with a modied water level.

The tk (n) term can be interpreted as a taxation term. Itlowers the water-lling level (and hence the transmit powerspectrum density) in frequencies where the effect of its inter-ference to other users is strong. This is done on a tone-by-tonebasis. The outer loop sets the value of the taxation term. Eachuser responds by maximizing the weighted sum rate over itsown power allocation P k (n). The weighting factor k also hasan effect of adjusting the overall water-lling level.

As compared to previous work, the proposed algorithm hasthe following main advantages:

In previous work, the update of k is always done via asubgradient method [4] [7] [8], which can be slow. Theproposed bisection method is much more efcient.

The proposed algorithm has an interpretation of beinga modied water-lling process for each user as shownin Fig. 3. This facilitates distributed implementation, astk (n) is the only information that needs to be exchangedamong the users, while the interference term I k (n) canbe measured by each modem in real time. In addition, amodication of the water-lling level is the only changethat is required in implementation. This facilitates theimplementation of discrete bit-loading.

The modied iterative water-lling algorithm can beeasily adapted to asynchronous DSL systems with in-tercarrier interference [6]. This is a key advantage ascompared to OSB [4] or its derivatives [5] [9], and otherpreviously proposed algorithm such as SCALE [10]. Toaccount for intercarrier interference, the crosstalk term(20) only needs to be modied as (16). The taxation term(19) only needs to be modied as shown in (25) at the topof the next page 4, while all other parts of the algorithmremain the same.

As typical in iterative-water-lling-like algorithms, the con-vergence proof is difcult to establish in full generality. Itis possible to adopt an approach of [7] to show that undera weak-interference condition, the iterative modied water-lling inner loop for a xed t k (n) would converge. However,

4In (25), SINR j (r ) is dened to be the signal-to-interference-and-noiseratio for j th user at the r th tone.

12k feet

3k feet

CO

Optical Fiber

RT12k feet

Fig. 4. Loop topology for two downstream ADSL users. User one is deployedfrom the central ofce (CO). User two is deployed from a remote terminal(RT). The RT user emits strong crosstalk to the CO user.

0 1 2 3 4 5 60

1

2

3

4

5

6

CO Downstream Rate (Mbps)

R T D o w n s

t r e a m

R a

t e ( M b p s

)

upper bound

proposed algorithm

OSBICI

greedy bitadding

greedy bit subtracting

iterative waterfilling

Fig. 5. Comparison between rate regions of the proposed algorithm anditerative water-lling [3], OSB [4], and two greedy algorithms proposed in[6] for the two-downstream-ADSL scenario with intercarrier interference. TheOSB performs poorly because the optimal spectrum is derived assuming noICI.

in practice, these weak-interference conditions are not alwaysmet, yet convergence is still observed.

The convergence of t k (n) in the outer loop is also not easyto establish. However, one may ensure convergence by slowingdown the update of tk (n) by incorporating a memory term.For example, let tk (n) be the taxation term as computed by(19), the following update rule can be used to slow down theupdate

tnewk (n) = toldk (n) + (1 )tk (n). (26)

for some 0 < < 1.Finally, we note that since the proposed algorithm solves the

KKT system directly. When it converges, it always convergesto a KKT point, which is guaranteed to be at least a localoptimum solution to the original problem (15).

Note that the proposed algorithm reduces to the originaliterative water-lling algorithm [3] when the taxation terms areset to be tk (n) = 0 . Thus, the proposed algorithm is attractivefrom an implementation point of view, as it allows legacysystems to be upgraded gradually.


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tk (n) =j = k

j |H kj (n)|2N 2 1

r = N 2

P j (r )|H jj (r )|2

P j (r )|H jj (r )|2 + l= jN 2 1

m = N 2 (m)P l (r m)|H lj (r m)|2 + 2

(r n)

l= j

N 2 1

m = N 2 (m)P l (r m)|H lj (r m)|2 + 2

(24)

=j = k

j |H kj (n)|2

N 2 1

r = N 2

[SINRj (r )]2

1 + SINR j (r ) (r n)P j (r )|H jj (r )|2 (25)

V. N UMERICAL S IMULATION

In this section, we present a simulation example to illustratethe effectiveness of the proposed modied iterative water-lling algorithm. We choose a particularly strong crosstalk scenario in asymmetric digital subscriber lines (ADSL) asshown in Fig. 4, where the line deployed from a remoteterminal (RT) emits a strong crosstalk signal to the line

deployed from a central ofce (CO). In addition, intercarrierinterference is assumed to be present.The achievable rate region for the proposed algorithm as

compared to the iterative water-lling algorithm [3], the OSBalgorithm [4], and two greedy algorithms proposed in [6]is presented in Fig. 5. As shown, because of the presenceof ICI, the OSB algorithm performs rather poorly. This isexpected as OSB optimizes spectrum assuming no ICI, andthe optimized spectrum typically involves frequency-divisionmultiplexing, which leads to poor performance when ICI ispresent. The iterative water-lling algorithm also does notperform particularly well, because it does not take into accountthe effect of strong interference from the RT user to the CO

user. The two greedy algorithms proposed in [6] are basedon local greedy subtraction or addition of bits starting fromsome initial spectrum. While outperforming OSB and iterativewater-lling, they still do not perform particularly well. Theproposed algorithm produces a larger achievable rate than allprevious algorithms. In Fig. 5, an outer bound, which is com-puted without the ICI terms, is also plotted. Clearly, the outerbound is not achievable. Although the proposed algorithm onlyachieves local optimal solutions, the local optimum obtainedappears to be fairly close to the global optimum for thisparticular example. Finally, we note that simulation experienceshows that the proposed algorithm is also more efcient thanOSB and the greedy algorithms proposed in [6].

VI . C ONCLUDING R EMARKS

While the performance of a single-user system is notnecessarily a very sensitive function of the input spectrum,the performances of multiuser systems often are becauseof the presence of interference. For this reason, spectrumoptimization has become an important part of the overallcommunications system design in many practical multiusersystems.

However, spectrum optimization in the presence of crosstalk interference is a challenging problem. For a general multiuserchannel setting, the problem formulation leads to nonconvexstructures which are often analytical intractable and numer-ically difcult to solve. This paper investigated two specicexamples of practical importance in which either analytic orefcient numerical solutions are available. Interestingly, bothsolutions resembles a single-user water-lling solution, except

with a modied water-lling level. These single-user-water-lling-like solutions are attractive, because they are efcient,robust, and are often amendable to distributed implementation.

ACKNOWLEDGMENT

This work was supported in part by the Natural Scienceand Engineering Research Council of Canada, and in part byHuawei Technologies, China. The author wishes to thank Mr.Jun Yuan for performing the simulation in the paper.

REFERENCES

[1] J. M. Ciof, G. P. Dudevoir, M. V. Eyuboglu, and G. D. Forney, MMSEdecision feedback equalizers and coding: Part I and II, IEEE Trans.Commun. , vol. 43, no. 10, pp. 25822604, Oct. 1995.

[2] D. P. Palomar, J. M. Ciof, and M. A. Lagunas, Joint Tx-Rx beam-forming design for multicarrier mimo channels: A unied framework for convex optimization, IEEE Trans. Signal Processing , vol. 51, no.9, pp. 23812401, Nov. 2003.

[3] W. Yu, G. Ginis, and J. M. Ciof, Distributed multiuser power controlfor digital subscriber lines, IEEE J. Select. Areas Commun. , vol. 20,no. 5, pp. 11051115, June 2002.

[4] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen,Optimal multiuser spectrum balancing for digital subscriber lines, IEEE Trans. Commun. , vol. 54, no. 5, pp. 922933, May 2006.

[5] W. Yu and R. Lui, Dual methods for nonconvex spectrum optimizationof multicarrier systems, IEEE Trans. Commun. , vol. 54, no. 6, pp.13101322, June 2006.

[6] V. M. K. Chan and W. Yu, Multiuser spectrum optimization for discretemultitone systems with asynchronous crosstalk, IEEE Trans. SignalProcessing , 2006, submitted.

[7] J. Huang, R. Cendrillon, M. Chiang, and M. Moonen, Autonomousspectrum balancing (ASB) for frequency selective interference chan-nels, in IEEE Inter. Symp. Inf. Theory (ISIT) , Seattle, U.S.A., 2006.

[8] J. Yuan and W. Yu, Distributed cross-layer optimization of wirelesssensor networks: A game theoretic approach, in Global Telecommuni-cations Conf. (GLOBECOM) , San Francisco, U.S.A., 2006.

[9] R. Cendrillon and M. Moonen, Iterative spectrum balancing for digitalsubscriber lines, in IEEE Inter. Conf. Commun. (ICC) , Seoul, Korea,2005.

[10] J. Papandriopoulos and J. S. Evans, Low-complexity distributedalgorithms for spectrum balancing in multi-user DSL netwoks, in IEEE Inter. Conf. Commun. (ICC) , Istanbul, Turkey, 2006.

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